ECON 2310 Final Exam Review PDF

ECON*2310 Final Exam Review Chapter 2: SUPPLY AND DEMAND Demand Curve: shows how much buyers of the product want to buy at each possible price, holding fixed all other factors that affect demand (except the price of the good) - A change in the price of the good causes a movement along the demand curve. A change in some other factor of demand (such as consumer tastes, or income or the prices of other goods) causes a short of the demand curve - Two products are substitutes if, all else equal, an increase in the price of one of the products causes buyers to demand more of the other product. Or, products are substitutes, if consumers are willing to substitute one good for the other. - Two products are complements if, all else equal, an increase in the price of one product causes consumers to demand less of the other product. Or, products are complements if consumers only want to consumer the two products together. Supply Curve: shows how much sellers of a product want to sell at each possible price, holding fixed all other factors that affect supply (expect the price of a good) - A change in the price of the good causes a movement along the supply curve. A change in some other factor of supply (such as technology or input prices) causes a shift of the supply curve Market Equilibrium: market price and market quantity at which quantity demanded is equal to quantity supplied - To judge how large changes in market price and market quantity will be, we need a measure of the price responsiveness of supply and demand. However, this measure should not depend on the units of measurement (price per liter vs. price per 4-liter container) Price Elasticity of Demand: the percentage change in the amount of the good demanded for each one percent increase in the price ∆𝑄𝐷 𝑑 𝑄𝐷 𝐸 = ∆𝑃 𝑃 - For a linear demand curve: the price elasticity is different at different points on the curve. - For a linear demand curve we can figure out the demand curve if we know price, quantity and price elasticity - Ed is between -∞ (perfectly elastic demand) and 0 (perfectly inelastic demand) - Demand is elastic when Ed is less than -1 (i.e., between -∞ and -1). Demand is inelastic when Ed is larger than -1 (i.e. between -1 and 0) Total Expenditure and Elasticity of Demand: a small increase in price causes total expenditure to increase if demand is inelastic and to decrease if demand is elastic - Total expenditure for a linear demand curve is largest at the price at which Ed = -1 Price Elasticity of Supply: is the percentage change in the amount of the good supplied for each one percent increase in the price ∆𝑄𝑆 𝑆 𝑄𝑆 𝐸 = ∆𝑃 𝑃 - Es is between ∞ (perfectly elastic supply) and 0 (perfectly inelastic supply) - Supply is elastic when Es is larger than 1 (i.e between 1 and ∞). Supply is inelastic when Es is less than 1 (i.e between 0 and 1) Comparing Two Linear Curves: When two demand curves coincide at a particular price (i.e. they cross), then the steeper one is less elastic at that price (similarly for supply curves) When Supply Shifts: the less elastic the demand curve at the initial equilibrium price, the larger the price change and the smaller the change in market quantity When Demand Shifts: the less elastic the supply curve at the initial equilibrium price, the larger the price change and the smaller the change in market quantity Income Elasticity of Demand: is the percentage change in quantity demanded for each one percent increase in income (positive for normal goods, negative for inferior goods) 𝑑 ∆𝑄 𝑀 𝐸𝑀 = ∆𝑀 * 𝑄 Cross-Price Elasticity: is the percentage change in the quantity demanded of one good for each one percent increase in the price of the other good (positive for substitutes, negative for compliments 𝑑 ∆𝑄 𝑃𝑜 𝐸𝑃𝑜 = ∆𝑃𝑜 * 𝑄 Quiz 1 Questions 1. Suppose that the demand for corn in billion bushels is given by Qc = 20-4Pc+8Pp-O.5Pb. Butter costs $8 per kilogram and potatoes cost $0.5 per kilogram. At what price of corn will consumers demand 8 billion bushels? Answer: 3 Help: Plug in the prices of the other goods into the demand function. Then set the demand function equal to 8 and solve for the price of corn. 2. In the previous question, the price of butter negatively influences the demand for corn because butter and corn are: Answer: Compliments Help: Corn and butter are complements. They are usually eaten together. SO, if the pierce of butter goes up, less butter and therefore less corn is consumed 3. Suppose that demand for a given good is given by Qd = 30-P. Supply is given by Qs = 6+b*P, where b = 5.0. What is the equilibrium demand of the good? Answer: 26 4. A vertical supply curve represents a supply that is: Answer: perfectly inelastic 5. Suppose that the demand function for a good is given by Qd=30-P. Suppose that the price of the good is $20.00. What is the elasticity of demand at the price? Answer: -2 Help: The elasticity of a linear demand curve is easily calculated. It is equal to ∆𝑄𝐷 𝑑 𝐸 = 𝑄𝐷 ∆𝑃. Notice that the first term is equal to the slope of the demand function. 𝑃 Here that is -1. Then you calculate the quantity at the given price to calculate P/Q. Multiply P/Q with the slope and that gives you the elasticity. Important: If you have an inverse demand function, then the first term is not equal to the slope, but equal to 1/slope. Try this with a demand function that does not have a slope of 1. Check that you get the same elasticity at a certain price, no matter whether you calculate it from the demand or the inverse demand (taking into account that the first term of the elasticity is calculated differently). 6. Total Expenditure and the Elasticity of Demand: A small increase in price causes total expenditure to increase if demand is inelastic, and decrease if demand is elastic. Total expenditure is largest at the price for which the elasticity is equal to -1. Suppose that the demand function is given by Qd=24−b∗P. Suppose that b = 1.0. At which price is consumer expenditure maximized? Answer: 12 Help: You know the elasticity formula. So, you also know that it has to be true here that Ed=−1=−b∗(P/(24−b∗P)) Solve this equation for P. Notice: The price you calculated is the midpoint along your demand function. This is always true for a linear demand function: Expenditure is maximized at the midpoint of the demand function. 7. Insert larger or smaller in the blanks below: When the demand curve shifts: The less elastic the supply curve at the initial equilibrium price, the larger the price change and the smaller the change in the amount bought and sold. 8. When people consumer steak when income goes up (because now they can afford to buy steak rather than ramen noodles or Kraft dinner), then a steak is a: Answer: Normal Good Help: If steaks are a normal good, like in this question, then the income elasticity is positive 9. Please insert complements or substitutes and positive, negative or zero, and larger or smaller in the blanks below If people like to eat either corn or potatoes at a barbeque, then corn and potatoes are substitutes. In this case the cross-price elasticity is positive. The lager the price of potatoes, the larger the amount of corn that is bought. 10. The two supply curves represented by P = 1/2Qs and P = 20Qs have the same price elasticity. True or False. Answer: True 11. Refer to the previous question about total expenditure and the elasticity of demand. Check whether the following statement is true for the demand function in that question: Total revenue is maximized at the midpoint of the downward sloping linear demand curve. Answer: True Help: This statement is true for all linear demand curves: Total expenditure is maximized at the midpoint of the demand curve. This is also the point where the elasticity is equal to -1. 12. Suppose that the price elasticity is equal to -3 at a price of $5 and a quantity of 750 units. This information is consistent with the following demand function: Qd=2250-300P. True or False. Answer: False Help: First check whether the price elasticity is satisfied. If yes, then also check whether the point is really on the demand function (notice that the elasticity only used the slope of the demand function and not the intercept). 13. When the demand curve shifts, the change in equilibrium price will be smaller the more inelastic the supply curve. True or False. Answer: False 14. Along the linear demand curve, demand becomes more elastic as we move towards the price intercept. True or False. Answer: True 15. An increase in the price of corn results in a movement along the demand curve for corn. True or False. Answer: True Chapter 4: CONSUMER PREFERENCES Preferences: tells us about a consumer’s likes and dislikes (taste) - Tells us how a consumer ranks consumer bundles Consumption Bundles: are collections of goods and services that a consumer consumes The Ranking Principle: a consumer ranks, in order of preferences (though possible with ties), all potentially available alternative (preferences are complete and transitive) The More is Better Principle: When one consumption bundle contains more of every good than a second, then a consumer prefers the first bundle. (For every ‘bad’ there is a ‘good’ (work time/leisure time)) The Choice Principle: Among the available alternatives (possibly constrained by income, etc.), the consumer selects the one (or one of those, if indifferent) that he/she ranks highest. Indifference Curves: A convenient way to represent preferences graphically. An indifference curve connects the set of all bundles that have the same rank or that give the same level of well-being. An indifference curve connects all bundles that a consumer likes equally. - They are thin - When consumers like both goods, indifference curves do not slope upward - The indifference curves of a single consumer do not cross Special Indifference Curves: Perfect substitutes and perfect complements Marginal Rate of Substitution MRSXY: The rate at which a consumer must adjust good Y to maintain the same level of well-being when good X changes by a tiny amount from a given starting point. It is equal to the slope of the IC multiplied with -1 Declining MRS: An IC has a declining MRS if it becomes flatter when moving along the horizontal axis. IT has a declining MRS if MRS becomes smaller when good X increases. Utility and Utility Functions: The utility of a consumer summarizes everything that is known about a consumer’s preferences. Utility is a numeric value indicating relative well-being. Higher utility indicates higher satisfaction than lower utility. A utility function assigns a utility value to every consumption bundle. Module on CONSUMER PREFERENCES - To represent a consumer’s ranking of all possible consumption bundles, we draw indifference curves. Along an indifference curve, all bundles give the same utility - Indifference curves are thin and for the same consumer cannot cross. They are downward sloping if the consumer likes both goods. All of these can be shown using the More-Is-Better principle - How to figure out the shape of an indifference curve? Pick a random bundle. Think of another bundle that makes the consumer equally well off. - For example, pizza and coke (given that the consumer likes both): Take a bundle that has both pizza and coke. If I take some pizza away from the consumer, I need to give extra coke to the consumer so that the consumer is as well off as before. SO the new bundle lies to the south-east of the original one. THe IC is downward sloping - For example, consumption and labor: Take a bundle that has positive amounts of both consumption and labor. If I take some consumption away from the consumer, I also need to take some labor away. Why? Taking consumption away makes the consumer worse off. To compensate, I also need to reduce the (disliked) labor. The new bundle lies to the south-west of the original one. The IC is upward sloping. - Use the same procedure to figure out the shape for all other types of preferences - There are three different types of utility functions that we discussed in class (along with the respective ICs): - Declining MRS: ICs are downward-sloping curves. Cobb-Douglas functions are examples of those. They take the form U(x,y) A*xα*yβ. As the name suggest, the MRS of these ICs is declining along an indifference curve - Perfect Substitutes: ICs are straight, downward sloping lines. MRS is always constant. They take the form U(x,y) = ax+by - Perfect Complements: ICs are L-shaped. MRS is not defined at the corner. They take the from U(x,y) = min[ax, by] - In general, any utility function that is not perfect substitutes and not perfect complements must be declining MRS. As a rule of thumb, it may be easier to exclude the two special cases first. - The MRS tells us at which ratio a consumer is willing to give up one good for another while keeping utility constant. It answers the following question: If I give up one unit of x, how many units of y do I need in return. It can be calculated using the marginal utilities. For example, if MRSXY = MUx/MUy = 10, then the consumer needs 10 units of y in return for giving up one unit of x. If MRSXY = 1/10, then the consumer needs 1/10 units of y in return for giving up one unit of x. Or, in other words, the consumer needs 1 unit of y in return for 10 units of x. The MRS is equal to (-1)*slope of IC. So the IC is downward sloping, the MRS is positive - The marginal utilities MUx and MUy are derivatives of the utility function. That means that the different utility functions have different MUs. When working on a question, make sure you use the MUs given for that question and not MUs that you remember from class or the book Quiz 2 Questions: 1. Which of the following statements best describe Mia’s preferences (utility on I2 is higher than on I1)? Answer: Meat makes MIa neither happy nor unhappy. Bread however, she likes Help: From the indifference curves you can see: Given a certain amount of bread Mia’s utility does not change independent of how much meat she has. Therefore, Mia neither likes nor hates meat. Mia’s utility goes up with the amount of bread she has, so she likes bread. 2. Two indifference curves for the same individual may have equal utility. True or False. Answer: False Help: An indifference curve connects all the bundles that yield the same level of utility. Therefore, no two indifference curves can yield the same level of utility. 3. Esteban likes both chocolate ice cream and lemon sorbet. His preferences correspond to the utility function: U(C,S) = C1/3S2/3, where C stands for scoops of chocolate ice cream and S stands for scoops of lemon sorbet. Esteban would rather have four scoops of chocolate ice cream and two scoops of lemon sorbet than two scoops of chocolate ice cream and four scoops of lemon sorbet. True or False. Answer: False 4. Cindy has the preference given by U(x,y) = min[x,y]. Answer: Cindy prefers bundle (2,100) to bundle (1,1000) Help: The utility function representing perfect complements always takes the form U(x,y)=A*min[cx,dy], where Ac and d are positive numbers. 5. The utility function U = X+Y and U=X+Y+1 will both rank bundles of goods X and Y identically. True or False. Answer: True Help:While the absolute utility that the two functions assign to two bundles differ, the ranking of bundles is the same for both utility functions. 6. Molly’s utility function is: U(x,y) = y + 4√x. She has 36.0 units of x and 8.0 units of y. If her consumption of x is reduced to 0, how many units of y would she need in order to be exactly as well off as before? Answer: 32 Help: Suppose, for example, that she has 10 units of good y and 25 units of good x. Then her total utility from this bundle is U = 10+4*5 = 30. If she consumes 0 of x, then she needs 30 units of y, since each unit of y generates one unit of utility. 7. Indifference curves can only be upward sloping if one of the goods is a “bad”. True or False. Answer: True 8. John’s MRSBM for reading books with watching movies is three movies (per books) regardless of the amounts consumed Answer: He would rather read two books and watch no movies than read no books and watch two movies. Help: John’s preferences are perfect substitutes because his MRS is always the same independent of his current consumption bundle. Since he is willing to give up three movies for one book, he receives the same utility from three movies as he does from one book. Therefore, his utility function can be written as U(B,M) = 3B+M. The utility function describing perfect substitutes always takes the form U(x,y) = ax+by, where a and b are positive numbers. 9. Suppose Anne has 80 peanuts and is willing to exchange 5 peanuts for one walnut. Bob has 20 walnuts and is willing to exchange one walnut for 6 peanuts. Anne and Bob will trade peanuts for walnuts. True or False. Answer: False Help: Bob will only give up walnuts if he receives at least 6 peanuts per walnut. But Anne is not willing to make his trade. Also: Bob likes walnuts more than Anne (he needs to receive more peanuts in return), but he already has all the walnuts, so there is no trade that both would agree to. 10. Consumption bundles A, B and C all lie on a straight downward sloping line. The line is NOT an indifference curve. The consumer’s preferences has a declining MRS and the consumer likes both goods. Bundles A and B lie on the same indifference curve. Which of the following is correct? Answer: The consumer prefers C to A and B Help: The consumer is indifferent between A and B because they lie on the same indifference curve. As you can see from the diagram below, she prefers C to A and B because C is necessarily on a higher indifference curve. 11. Suppose that a consumer has the following utility function for chocolate ice cream and lemon sorbet: U(C,S) = C1/3S2/3. Then the marginal utilities are: MUc = S2/3/3C2/3 and MUs = 2C1/3/S1/3, respectively. Calculate the MRSCS when one scoop of chocolate ice cream and 9.0 scoops of lemon sorbet are chosen. Answer: 4.5 Help: Notice that a utility function of the form U(x,y) = (xα)(y1-α) is called Cobb-Douglas utility function. While the marginal utilities are complicated, the MRSXY simplifies quite a bit. For all utility functions of this form, it is true that MRSXY = a/(1-a) * y/x 12. If in the previous question the utility function was give by U(x,y) = (xα)(y1-α), the marginal utilities would be MUx a(y1-α)/(x1-α) and MUy = ((1-a)xα)/(yα). Calculate the value for the α in the utility function if the MRSCS = 3.00 and equal amounts of chocolate ice cream and lemon sorbet are chosen. Answer: 0.75 Help: This is also a Cobb-Douglas utility function, so the simplification for calculating the MRS applies as well. Also: Since the amounts of chocolate ice cream and lemon sorbet are equal they cancel out and we are left with MRSCS = a/(1-a). Chapter 5: CONSTRAINTS, CHOICES AND DEMAND Budget Constraints: identifies all consumption bundles a consumer can afford over a period of time Budget Line: all consumption bundles that just exhaust a consumer’s income 𝑀𝑈𝑥 𝑝 Tangency Condition: 𝑀𝑅𝑆𝑋𝑌 = 𝑀𝑈𝑦 = 𝑝𝑥 𝑦 𝑀𝑈𝑥 𝑀𝑈𝑦 ‘Bang for the Buck’: 𝑝𝑥 = 𝑝𝑦 Price Consumption Curve: shows how the best affordable consumption bundle changes as the price of a good changes, holding everything else fixed (preferences, income, other prices) Individual Demand Curve: describes the relationship between the price of a good and the amount a particular consumer purchases, holding everything else fixed Income Effect: change in consumption of a good that results from a change in income Income Consumption Curve: shows how the best affordable consumption bundle changes as income changes, holding everything else fixed Engel Curve: for a good describes the relationship between income and the amount consumed, holding everything else fixed - If the demand for a product increases when income increases the product is called a normal good. If instead it decreases, it is called an inferior good - The income elasticity of demand is positive for normal goods and negative for inferior goods - At least one good must be normal starting from any particular income level - No good can be inferior at all levels of income Module on CONSTRAINTS, CHOICES AND DEMAND, SAVING AND BORROWING - Consumers have a fixed income. To maximize their utility, they spend their entire income on the goods available in the economy - Budget Line: M = px*x + py*y - Along the budget line, the consumer spends their entire income - To figure out what a change in income M or prices do to the budget line, think about what it does to the slope. px/py (for example, smaller slope means flatter budget line) and/or to the intercepts (you can use numbers as an example in none are given) Utility Maximization/Optimal Choice - Given their income and preferences, consumers want to buy the highest indifference curve that their income allows them to buy (the optimal choice will always be on the budget line) - ‘Bang for the Buck’: As a simple rule, consumers always want to purchase the good that gives them greater ‘bang for the buck’ ( or in other words, the most extra utility per dollar spend on it) We can calculate the ‘bang for the buck’ for good i by calculating MUi/pi a) Declining MRS - Remember that for these ICs , the MRS changes along the IC - The optimal bundle is characterised by the fact that the slope of the IC and the slope of the budget line are equal: 𝑀𝑈𝑥 𝑝 𝑀𝑈𝑥 𝑀𝑈𝑦 𝑀𝑅𝑆𝑋𝑌 = 𝑀𝑈𝑦 = 𝑝𝑥 , 𝑝𝑥 = 𝑝𝑦 𝑦 - The bundle is optimal because both goods deliver the same ‘bang for the buck’ Calculating the Optimal Bundle for Declining MRS 𝑀𝑈𝑥 𝑝 - 𝑀𝑅𝑆𝑋𝑌 = 𝑀𝑈𝑦 = 𝑝𝑥. Solve this equation for either x or y 𝑦 - M= px*x + py*y. Replace either x or y with the expression from 1). Solve to find the amount of the other good, use expression from 1) again. b) Perfect Substitutes - 𝑀𝑅𝑆𝑋𝑌 = a/b. Either you remember this or you find a bundle on the x-axis and y-axis between which the consumer is indifferent. Remember that the MRS is always the same - That means that the slope of the indifference curve is always either larger, smaller or the same as the price ratio There are 3 cases: 𝑀𝑈𝑥 𝑀𝑈𝑦 - 𝑀𝑅𝑆𝑋𝑌 > 𝑝𝑥/𝑝𝑦 ↔ 𝑝𝑥 > 𝑝𝑦 ICs are steeper than the budget line. Consumers only buy good y. 𝑀𝑈𝑥 𝑀𝑈𝑦 - 𝑀𝑅𝑆𝑋𝑌 < 𝑝𝑥/𝑝𝑦 ↔ 𝑝𝑥 < 𝑝𝑦 ICs are flatter than the budget line. Consumers only buy good y. 𝑀𝑈𝑥 𝑀𝑈𝑦 - 𝑀𝑅𝑆𝑋𝑌 = 𝑝𝑥/𝑝𝑦 ↔ 𝑝𝑥 = 𝑝𝑦 ICs have the same slope as BL everywhere. Any bundle along the budget line is optimal. c) Perfect Complements. Good x and y always need to be consumed in a fixed proportion - The optimal bundle is always at the corner of an IC - At this corner U = ax = by. So we get an optimal ratio of goods x and y, i.e., x/y = b/a. We also write this as x = (b/a)*y and use this in the budget constraint to get the optimal bundle. Saving and Borrowing - It is just an optimal choice in which the two goods are co and c1 instead of x and y. Depending on the consumer’s preferences, we use the same procedure as above to find the optimal bundle - Only difference: the budget constraint. Mo and M1 are income today and tomorrow, respectively. R is the interest rate (e.g. R=0.5) 𝑐1 𝑀1 𝑐𝑜 + 1+𝑅 = 𝑀𝑜 + 1+𝑅 - If you know that the optimal bundle is such that the consumer consumes nothing tomorrow then c1 = 0 in the above budget line. Therefore you can calculate co - If the interest rate changes, the budget line rotates around the bundle (𝑀𝑜,𝑀1) - If 𝑐𝑜 < 𝑀𝑜, the consumer is a saver. - If 𝑐𝑜 > 𝑀𝑜, the consumer is a borrower Quiz 3 Questions 1. If both prices double and income triples, then the budget line rotates and shifts outward. True or False Answer: False Help: The slope of the new budget line is 2px/2py = 2/2 * px/py = px/py, sp it does not change. Income however, triples, so that the new budget line is then parallel to the old one but shifts outward 2. The price of bread is $7.50 per kilogram, and the price of butter is $15.0 per kilogram. Rupert buys 12 kilograms of bread and his income is $120. How much butter does he buy, assuming that he consumes nothing else than butter and bread? Answer: 2 Help: At the current prices, Rupert spends $90 on bread (he buys 12 kilograms at a price of $7.50 per kilogram). Therefore, he spends the remaining $30 on butter. Given the price of butter, you can figure out how many kilograms of butter he can buy with $30. 3. Alan can spend $10 a week on snacks. He likes ice cream, which costs $1 per scoop and popcorn which costs 40 cents per bag. All of a sudden, however, his sister Alice starts stealing half of his popcorn every time he buys any. Which of the following statements is true about Alan’s budget constraint in a diagram with popcorn on the x-axis. Answer: It rotates around the intercept on the y-axis Help: If Alan’s sister steals half of his popcorn, then effectively the price of the popcorn has gone up to $0.80 per bag; Alan has to buy two bags to consume just one. The constraint rotates around the intercept on the y-axis. 4. Clara spends her entire budget and consumes 8.0 units of x and 13 units of y. Her income doubles and the price of y doubles, but the price of x stays the same. If she continues to buy 13 units of y, what is the largest number of units of x that she can afford? No units, no rounding. Answer: 16 Help: If the price of y doubles but Clara consumes the same amount now as before the price change, then her expenditure on y doubles. Income, however, doubles as well. So, the share of income spent on y is the same as before. Therefore, the share of income spent on x is also the same as before. However, if the share of income spent on x is the same, but income doubles, Clara can buy twice as many x (since the price of x does not change). 5. Suppose Sam and Sara both consider movies and books to be perfect substitutes. If they both face the same prices, it cannot be true that Sam consumes only books while Sara consumes only movies.Choose the appropriate statement. True, it cannot be the case. False; it is possible that Sara consumes only movies while Sam consumes only books. Answer: False Help: Yes, it is possible that they consume different optimal bundles, even if they face the same prices. Suppose that Sam is a fast reader, so he finishes a book within the same time it would take him to watch a movie (he is a REALLY fast reader). So, he is indifferent between one movie and one book. Sara, on the other hand, likes to take her time with reading. So, if she wants to be entertained for the same period of time, she is indifferent between one book and six movies. Now suppose that books are on sale: One book costs $5 and one movie costs $10. A bundle with one book costs Sam $5, while a bundle with one movie costs him $10. Since he is indifferent, he will choose the relatively cheaper good: Books. For Sara, it's different: A bundle with one movie costs $10, while a bundle with six movies costs $60. Since she is indifferent between watching six movies and reading one book, she will choose the good that is relatively cheaper for her: Movies. Note that this is always the case: People with perfect-substitute preferences will choose only one of the two goods in their optimal consumption bundle. And they will choose the one that is relatively cheaper for them. Which one that is depends on the prices and their MRS. An easy way to check is to use the bang-for the-buck condition: People with perfect-substitute preferences will only consume the good for which the bang for the buck is larger. 6. Assume that Jennifer consumes her optimal bundle (X*,Y*). Furthermore we know that at that bundle MUX /PX > MUY /PY. It must be true that Y*=0, i.e. she consumes zero units of good Y. True or false. Answer: True Help: At Jennifer's optimal bundle, the 'bang for the buck' is greater for good x. She'd like to spend even more money on good x, but can't (after all it's her optimal bundle). So, she must already be spending her entire income on x. So, y*=0. 7. Olivia has received a $15 gift certificate that is redeemable only for roasted peanuts. Bags of roasted peanuts come in two sizes, regular and jumbo. A regular bag contains 30 peanuts and a jumbo bag contains 90. Olivia cares only about the number of peanuts she has, not about the size of the bag or the number of bags she needs to purchase. A regular bag costs $0.5 and a jumbo bag costs $1.60. Which of the following statements is true about Olivia's optimal choice? Answer: Olivia will only buy regular bags Help: Olivia is indifferent between buying one jumbo bag and three regular bags. However, jumbo bags are 3.2 times as expensive as regular bags. One jumbo bag costs $1.6. However, the equivalent 3 regular bags together cost only $1.5. So, just as in Question 5, Olivia, whose preferences are clearly perfect substitutes, buys only the goods that are relatively cheaper for her: Regular bags. In general: Olivia's preferences can be represented by the utility function U=R+3J. When the budget line is steeper than her MRS_RJ=1/3, then she will buy only jumbo bags. When the budget line is flatter than her MRS_RJ, then she will only buy regular bags. And when the slope of the budget line and her MRS are equal. She may buy any bundle along the budget line. 8. Ashley spends all her income on gasoline and food. At first she earns $100, buys 25 litres of gasoline at $2 per litre, and purchases 10 kilograms of food at $5 per kilogram. Her income later rises to $200, but the price of gasoline increases to $5 per litre, and the price of food rises to $7 per kilogram. Which of the following statements is true? Answer: Ashley is better off than worse off Help: While we can indeed not determine her optimal bundle after the changes, we can still say that Ashley must be better off. Ashley's circumstances are described in the below diagram. Since her initial optimal bundle is still affordable at the new prices and income, you can see that there must be bundles that she can afford now that yield higher utility than her initial optimal bundle (for example those that satisfy more-is-better compared to her initial optimal bundle). Therefore, Ashley must be better off than before because she can buy a bundle that yields higher utility. 9. Assume that Roger's indifference curves have a declining MRS. Roger is contemplating spending his entire income on ten kilograms of chips and ten kilograms of nuts. At this particular bundle, his marginal utility from one extra kilo of nuts is 3 and his marginal utility from one extra kilo of chips is 4. The price of a kilo of nuts is $6 and that of a kilo of chips is $8. Roger should purchase fewer chips and more nuts if he wishes to maximize satisfaction. True or False. Answer: False Help: At Roger's consumption bundle, MUX /PX = MUY /PY. Therefore, both goods deliver equal 'bang for the buck'. Roger cannot increase his satisfaction by increasing the amount of one good and decreasing the amount of the other. Notice that Roger is already spending all his income. Therefore, he cannot increase the amount of one good without decreasing the other. Graphically, we can answer such a question as follows: Consider the diagram below: In this case, at the current choice, the MRS_CN < p_C/p_N. So, the budget line is steeper than the indifference curve. Along the BL to the northwest of the present bundle are bundles that are affordable and lie on a higher indifference curve. To reach these, Roger must buy fewer chips and more nuts. If, on the other hand, the MRS were larger than the BL, then the BL would be flatter than the indifference curve and Roger would have to buy less nuts and more chips to increase happiness. 10. Susan’s utility function is u(x,y) = xy. (MUx = y; MUy = x). Her income is 44.0 and px = 1 and py = 3. How many units of good x will she choose? Answer: 22 Help: We have seen a utility function like this before. This utility has a declining MRS. Therefore, you can use the recipe to find the optimal bundle. The optimal bundle satisfies the condition MRSXY=px/py. In addition we now that the budget constraint is met: M=pxx+pyy. This gives us two equations and two unknowns. Solving these equations for x yields the solution. Let us look at an example (recall that numbers are drawn randomly, so your numbers may be different than the ones in this example): Suppose that Susan's income is M=80. Let's use the recipe learned in class. Step 1: MRSxy = MUx/MUy = px/py. Fill in the information from the question, and we get, with MUs and prices: y/x=1/3→y=x/3. Step 2: Use the budget line. M=px∗x+Py∗y→80=1x+3y. Step 3: Substitute the solution from Step 1 into Step 2: 80 = 1x+3(x/3) = 1x+1x = 2x.→80 = 2x. So, we get x∗ = 40,y∗ = 40/3. Btw: For a Cobb-Douglas utility function, it is always true that MRSxy =α /β∗y/x. So, if you equate this to the price ratio to calculate the optimal bundle, you get α∗y∗py=β∗x∗px. Suppose that α=β. Then the consumer spends equal shares of her income on goods x and y since x*px =y *py. 11. Janet consumes two goods x and y. Her utility function is u(x,y) = x+2y (MUx = 1; MUy = 2). Her optimal choice is to buy 11.0 units of gd x and 20 units of good y. The price of good x is $1. HOw much is Janet’s income? Answer: 51 Help: Janet's preferences are perfect substitutes. Since she consumes positive amounts of both, good x and good y, it must be the case that the slope of the indifference curves (MRS) is identical to the slope of the budget line. Since you know the MUs, you can calculate the MRS. The price of good y must be equal to 2. Now you can calculate how much Janet spends on her optimal bundle. Since her bundle is optimal, her expenditure is equal to her income. 12. Andrew has preference given by u(x,y) = min{2x,3y}. Also px = py = 4.0. At his optimal consumption bundle, he achieves a utility of 90. What is Andrew’s income? Answer: 300 Help: Andrew’s preferences are an example of perfect complements. He will never spend more on one good than dictate by complementarity. So we know that he must consume at a corner of his L-shaped indifference curves, where 2x = 3y = 90. Thus he must consume 45 units of x and 30 units of y SO you can calculate the income given the prices of x and y. 13. Angela cares about consumption today (c0) and consumption tomorrow (c1). Assume that her MRS01 is c1/c0. Suppose that the price of consumption today and tomorrow are both equal to 1. Also assume that Angela earns $60 this year and $ 90.0 next year. The interest rate is $0.5 (or 50 percent). What is today's consumption (no rounding, no units)? Answer: 60 Help: Suppose, for example, that Angela's income today is 50 and her income tomorrow is 105. Plug those values into the following budget constraint 𝑐1 𝑀1 𝑐1 105 𝑐𝑜 + 1+𝑅 = 𝑀𝑜 + 1+𝑅 or 𝑐𝑜 + 1+𝑅 = 50 + 1+𝑅 We also know that MRSo1 = c1/co. At the optimal bundle, MRS01 = (1+R). So, c1=1.5*c0. Plug this relationship into the budget constraint and you get 180 = 2∗1.5∗co. For our example, this implies that c0 = 60. In order to achieve this, Angela must borrow $10 today because her income is only $50. So, tomorrow she must pay back $15. That leaves her $90 for consumption tomorrow, so that her consumption tomorrow is 90. CHAPTER 6: Demand to Welfare Dissection the Effects of a Price Change of One Good 1) That good becomes more expensive relative to all other goods. Consumers tend to shift their purchases alway from the more expensive good and towards other goods (substitution effect) - The substitution effect usually makes consumers buy less of a good that became more expensive (and never more of the good) - The substitution effect is usually negative for a price increase and positive for a price decrease, but it can be zero for certain preferences under certain circumstances 2) Consumers’ purchasing power declines. A dollar does not buy as much as it once did. Because consumers can no longer afford the consumption bundles they would have chosen if the price has not risen, they are effectively poorer and must adjust their purchases accordingly (income effect) - If a good is normal, the income effect reinforces the substitution effect. It is negative for a pierce increase and positive for a price decrease - If a good is inferior, the income effect opposes the substitution effect. It is positive for a price increase and negative for a price reduction - The income effect can be zero for certain preferences under certain circumstances Compensating Variation - It is the amount of money that exactly compensates a consumer for a change in circumstances - Allows us to put a monetary value on changes in well-being - Is negative when the consumer is better off under the new circumstances (e.g. price decrease) and positive if the consumer is worse off under the new circumstances (e.g. price increase) Consumer Surplus - Is the net benefit a consumer receives from participating in the market for some good - Is less exact tha the compensating variation in measuring changes in well-being but is sufficiently accurate for most purposes (if income effect is sufficiently small) Module on Demand and Welfare (Ch. 6) Income and Substitution Effects: the price of good x changes. The consumer adjusts demand for x. We imagine that this adjustment princess has two steps: Step 1) Substitution effect caused by the change in relative prices; Step 2) Income effect caused by the change in purchasing power Substitution Effect: Keep utility constant (same IC) but change relative prices. WHat is the consumer’s optimal bundle? We call this bundle the ‘substitution bundle’. The consumer needs more money than he/she has to buy the substitution effect bundle Income Effect: Keep relative prices constant but change purchasing power back to original income. What is the consumer’s optimal bundle now? Here: A diagram of substitution effect and income effect for declining MRS when the price of x goes up. A is the optimal bundle before the price change, B is the SE bundle, and C is the optimal bundle after the price change a) Declining MRS 1. A and C are just optimal bundles with the same income but different prices. Refer to the previous module for calculating an optimal bundle 2. The SE bundle also satisfies two conditions: a. MRS = new price ratio b. U(bundle A) = U(bundle B) b) Perfect Complements - The substitution effect is always zero - Total Effect = Income Effect c) Perfect Substitutes Remember that the optimal bundle is most likely a corner solution 1. MRS flatter than initial and new BL: Consumer buys only y before and after the price change. So, IE = SE = Total Effect = 0 2. MRS flatter than initial BL, but steeper than new budget line (only possible for a price decrease): Initially, the consumer buys only good y. After the price change, the consumer buys only good x 3. MRS same as initial or final BL: We cannot determine IE and SE because any bundle along the respective Budget Line is an optimal bundle 4. MRS steeper than initial and final budget line: consumer only buys x, before after the price change (This case can also occur for a price decrease. The the diagram would look slightly different) SE = 0. Total effect = income effect Some General Comments Substitution Effect: - The substitution effect usually make the consumer buy less of a good that becomes more expensive (and never more of that good) - The substitution effect is usually negative for a price increase and positive for a price decrease, but it can be zero for certain preferences/under certain circumstances (see cases above) Income Effect: - For a normal good, the income effect reinforces the substitution effect. It is negative for price increase and positive for a price decrease - For an inferior good, the income effect cancels out part of the substitution effect. It is positive for a price increase and negative for a price decrease. Generally, through, the size (Magnitude) of the substitution effect, os that demand is downward sloping even for inferior goods - The income effect can be zero for certain preferences/under certain circumstances (see cases above) A Few Applications of Income Effect and Substitution Effect - IE for most goods is small. Even for inferior goods the demand function is downward sloping (recall the difference between the demand function and the Engel curve). Given goods are the one exception: They are inferior goods with such a strong IE that the demand curve slopes upward - Backward bending labor supply: Choice over consumption and leisure. For a wage increase, SE for leisure is negative (Less leisure). However, IE is positive that leisure is a normal good (more leisure). Empirically, IE outweighs SE for large wages. We get a backward bending labor supply Measures of Consumer Wellbeing: a) Compensating Variation: Suppose we changed the consumer’s income to compensate for the change in wellbeing. How much money do we need to give/take away so that the consumer can purchase the same utility (i.e. a bundle on the same IC) as before? It is equal to the difference between the amount of money needed to purchase the SE bundle and the actual income of the consumer The compensating variation is the amount of money that exactly compensates a consumer for a change in circumstances. It therefore allows us to assign a monetary value on the utility change that the consumer experiences. IT is negative when the consumer is better off under the new circumstances (e.g. a price decrease) and positive if the consumer is worse off under the new circumstances (e.g. price increase) When calculating the compensating variation remember that it is the smallest subsidy that just compensates the consumer for the change in wellbeing. The consumer must be exactly equally well off as under the previous circumstances b) Consumer Surplus: Difference between willingness to pay for the good (measure by demand function) and market price. FOr linear demand functions. Triangle areas between demand function and price. Quiz 4 Questions 1. Which of the following statements is true? The income elasticity of demand is negative for normal goods The income elasticity of demand is positive for normal goods The income elasticity of demand is positive for inferior goods The income elasticity of demand is zero for inferior goods Answer: The income elasticity of demand is positive for normal goods Help: For normal goods, an increase (decrease) in income increases (decreases) the demand for the good. Therefore, the income elasticity is positive. 2. If the substitution effect and the income effect have the opposite sign, then the good cannot be a normal good. True or False Answer: True Help: COnsider a price increase. In this case, the SE is negative (it makes the consumer buy less of the good). If the IE has the opposite sign the IE must be positive. This means that for a decrease in purchasing power (resulting from the price increase), the consumer buys more of the goods. This is true for an inferior good, not for a normal good. A similar argument applies to a price decrease. 3. Suppose that a consumer’s indifference curves have a declining MRS. SUppose that the price of a good increases, but the consumer’s demand for the good does not change. Which of the following statements must be true in this case? Answer: The magnitude of the IE is the same as the magnitude of the SE and they have opposite signs Help: If the consumer's indifference curves have a declining MRS, then the substitution effect has to be negative. If the total change in demand is zero, however, then the income effect has to completely offset the substitution effect. It has to have the same magnitude and must be positive. 4. Bert's income is $120. He spends his entire income on books. Bert considers electronic books and paperback books to be perfect substitutes, and he is always indifferent between one paperback book and one electronic book. Initially, the price of a paperback book is $12 and the price of an electronic book is $10. Then the price of a paperback book increases to $15. Which of the following statements is the most accurate? Answer: Both SE and IE for paperback books are zero Help: Initially, Bert buys only electronic books. They cost $10 per book and Bert is indifferent between one electronic and one paperback book. After the price change, Bert still buys only electronic books. So, income and substitution effects for paperback books are both zero because his demand for paperback books does not change. It remains at zero. 5. Emily's income is $120. She spends her entire income on books. Emily considers paperback and electronic books to be perfect substitutes, and she is always indifferent between one paperback book and one electronic book. Initially, the price of a paperback book is $15 and the price of an electronic book is $10. Then the price of a paperback book falls to $8. Which of the following statements is true? Answer: For paperback books, the IE is 3 and the SE is 12 Help: Initially, Emily buys only electronic books. They are cheaper and she is indifferent between one electronic and one paperback book. With her income, she can afford 12 paperback books. Then the price of a paperback book falls to $8. In the final bundle, Emily buys only paperback books because they are not cheaper than electronic books. Therefore, the total change in demand is 15. The substitution effect bundle is on the same indifference curve as the initial optimal bundle, but at a point that is optimal given the new price ratio. Therefore, the substitution effect bundle contains zero electronic books and 12 paperback books. (Notice that this implies that the compensating variation is -24 because the consumer needs only $96 to purchase the substitution effect bundle.) Therefore, the substitution effect is 12 and the income effect is 3. 6. Consider a consumer with preferences u(x,y) = xy, (MUx = y, MUY = x) facing prices px = 2 and py = 1 and having an income of M = 24. Assume now that the price of good x increases to p’x = 4.5. Calculate the SED of the demand of good x associated with this price change. Answer: -2 Help: Indifference curves for these preferences have a declining MRS. Therefore you can use the recipe from class to calculate income and substitution effects. Recall that the substitution effect bundle must satisfy 2 conditions: At this bundle, the MRS is equal to the new price ratio. And the utility of this bundle is the same as the utility of the original bundle purchased before the price change. Notice that you do not need to calculate the final optimal bundle to answer the question. Since numbers are drawn randomly, here is a particular example to show how to solve the question. Suppose that px=2 and py=1. Income is M=24. Then the price of x increases to px=8. 1) What is the optimal bundle before the price change? a) MRSxy=px/py implies y/x = 2 and therefore y=2x. b) Plug this into the budget constraint: M=24=2x+y=2x+2x. This implies that x=6 and y=12. Btw, the utility at this bundle, which we will need later for the substitution effect bundle, is U=xy=6*12=72. 2) What is the 'substitution effect bundle'? a) At this bundle, the MRS is equal to the new price ratio, so that MRSxy=y/x=8. Therefore, y=8x. b) This bundle lies on the same indifference curve as the initial bundle. Therefore, the utility of the substitution effect bundle needs to be the same as the utility of the initial bundle. Above, we calculated this utility to be 72. Therefore, for the utility of the substitution effect bundle needs to be 72=xy. Plug the condition from part a into this and you get 72=x*8x. Therefore, x=3 and y=24. This is the substitution effect bundle. 3) What is the optimal bundle after the budget constraint? a) The price ratio is now the new price ratio, so MRSxy=y/x=8. (This is the same as condition a for the substitution effect bundle.) Therefore, y=8x. b) Plug this condition into the budget constraint: M=24=8x+y=8x+8x. Therefore, x=1.5 and y=12. The substitution effect is change in demand from the initial to the substitution effect bundle, so 3-6=-3. The income effect is the change in demand from the substitution effect bundle to the final bundle, so 1.5=3=-1.5. 7. Consider a consumer with preferences u(x,y)= 2√x + y, (MUx=1/x,MUy=1). These preferences have a declining MRSxy Assume that px=1 and py=1 and that the consumer has an income M=100. Calculate the income effect associated with good x if the price of good x increases to p'x=2.0. Answer: 0 Help: Since these preferences have a declining MRS, you can use the recipe from class to calculate optimal bundles. When you set the MRS equal to the price ratio, you must realize that the demand for good x is completely determined by this condition only. This implies that the demand for good x only depends on the price ratio and not on income. (To calculate the optimal amount of good y, calculate how much income you have left over to spend on y and how many units you can buy with that.) Once you realize this, you must also realize that the demand for good x in the substitution effect bundle and the final bundle must be the same. Therefore, the income effect is zero. The statements above are always true for these types of preferences. They are special preferences and are called quasilinear preferences. They always have the form U(x,y)=v(x)+y where v(x) is a function of x. The indifference curves of this utility function all have the same shape and are just vertically shifted versions of one another. A particular property of this type of preferences is that the amount bought of good x for a given price does not depend on the income. Therefore, the income-consumption curve is actually a vertical line. Given this, it makes sense that there is no income effect for good x. The total change in demand is caused entirely by the substitution effect. 8. Consider a consumer in an economy with two goods, x and y. Prices are px=3 and py=1. Preferences of this consumer are such that for any possible income and given the prices above, the following is true at any bundle: MUx/px>MUy/py. Assume now that the income of this consumer increases by $30.0. Calculate the change in demand for good x associated with this change in income. Answer: 10 Help: Since the marginal utility of good x per dollar spent on x is always larger than the marginal utility of good y per dollar spent, we know that this consumer will only consume good x. The change in demand is then simply the change in income divided by the price of x. 9. Assume an economy with two goods x and y. A consumer’s demand for good x is given by x = M/(px+py). Both goods x and y are normal goods. True or False. Answer: True Help: The demand function above gives the optimal amount of good x bought at certain prices and income. The consumer will therefore spend the rest of his or her income on good y. You can solve for the demand for good y by plugging the demand for x into the budget constraint and solving for y. You will find that the demand for y has the same form as the demand for x in this particular example. This implies that both goods are normal goods because the demand for both goods increases as income increases. 10. Consider an economy with two goods x and y. Assume a consumer has preferences: u(x,y)=xy.The consumer's optimal bundle consists of positive amounts of good x and good y. Prices are px=6.0 and py=1. What is the marginal rate of substitution (MRSxy) of this consumer at his or her optimal bundle? Answer: 6 Help: We know that these preferences have a declining MRS (we have seen them many times before). This means that if this consumer consumes positive amounts of both goods, it must be the case that MRS equals the price ratio. We know prices, so we can infer the MRS. If all consumers have preferences with a declining MRS (independent of the particular utility function), they will all consume optimal bundles at which the MRS is equal to the price ratio. Therefore, all consumers will consume bundles with the same MRS. That is also why no more trade between consumers will happen. 11. Assume an economy with two goods, x and y. A consumer has preferences u(x,y)=2(√x + √y),(MUx=1/√x,MUy=1/√y). Prices are px=1 and py=1. The consumer has an income of M=117.0. Calculate the CV (Compensating Variation) if the price of good x increases to px'=2. Answer: 39 Help: When you calculate the 'substitution effect bundle', you can calculate how much money the consumer would need to afford that bundle at the new prices. The compensating variation is the difference between that amount and the actual income of the consumer. Since the price of the good increases, the consumer needs to be compensated with a positive amount of money, so the compensating variation is positive. Here is an example. Income is random, so your answer may differ. Suppose that the consumer's income is $99. Step 1: Find the optimal bundle before the price increase using the recipe for declining MRS. We have done this multiple times, so I will omit this step. The answer is x*=y*=49.5. Step 2: Calculate the utility of the optimal bundle before the tax.U=2(√49.5 + √49.5)=4√49.5 Because of the hint in the question, I am not rounding this value.Step 3: The substitution effect bundle has the same utility as the original bundle, but is at a point where the indifference curve is tangent to a budget line with the new prices. So, we have to adjust our recipe:Step 3a: MRSxy=MUx/MUy=px/py √y/√x = 2/1 →y=4x Use this relation and plug it into the utility function Step 3b: U=4(√49.5)=2(√x+√y)=2(√x+√4x)=2(√x+√2x)=6√x 4√49.5=6√x 16∗49.5=36x x∗=22,y∗=88 Step 4: How much does this bundle cost at the new prices? Cost=22*2+88=132.Step 5: Compensation = cost-income = 132-99=33.On a side note: How much is tax revenue? For this, we would have to calculate the optimal bundle after the tax. For this bundle, we can use the usual recipe for declining MRS with the new prices and the consumer's income of $99. The final optimal bundle is x*=16.5 and y*=66. The tax is collected on good x. The per-unit tax is $1. So, tax revenue is $16.5. Compare this to the compensating variation. Once again, we have compensating variation > tax revenue. 12. Assume an economy with two goods, x and y. A consumer has preferences u(x,y) = xy, (MUx = y and MUy = x). Prices are px = 1 and py = 1. The consumer has an income of M = 140.0. Suppose that the price of good x increases to px’ =4, and that is due to a tax of 3 per unit of goods sold. Calculate the compensation variable and the tax revenue. WHat is the difference between CV and tax? Answer: 87.5 Help: Refer to the previous question to see how to get the compensating variation. When you compare compensating variation and tax, the compensating variation is much larger. That means that taxation is a 'leaky bucket': The utility lost by imposing a tax (measured by the compensating variation) is much larger than the tax that is collected. This means that the tax is inefficient! 13. Molly enjoys writing on paper. Currently, the price for 100 sheets of paper is $2.0. Molly's demand for paper is given by QP = 525 − 5000∗Pp where QP stands for her demand for sheets and PP stands for the price of a single sheet of paper. The government now imposes a tax of $0.50 on each pack of paper with 100 sheets. Therefore, the price of 100 sheets of paper rises to $2.50. Notice that Molly does not have to buy 100 sheets at a time. She can purchase individual sheets. Calculate Molly's change in consumer surplus (final CS-initial CS). Answer: -2.06 Help: Before the tax policy, the price of a single sheet of paper is $0.02. At this price, Molly purchases 425 sheets of paper. At this price, Molly's consumer surplus is equal to 0.5*(0.105-0.02)*425 = 18.06. After the tax policy, a single sheet now costs 0.025. At this price, Molly purchases 400 sheets and her consumer surplus is 05*(0.105-0.025)*400=16. Therefore, the change in consumer surplus is -2.06. 14. Calculate the tax revenue that the government collects from Molly in the previous question. Which of the following statements is true? Answer: The magnitude of the tax revenue collected is smaller than the magnitude of the loss in consumer surplus that MOlly experiences Help: After the implementation of the tax, Molly purchases 400 sheets of paper. The government collects 4*$0.5=$2 in taxes. The magnitude of the tax revenue is smaller than the magnitude of the loss in consumer surplus.This is an example in which we use the imprecise consumer surplus instead of the compensating variation to measure the loss in consumer wellbeing. Consumer surplus is easier to calculate because we only need information about the demand function and not the precise preferences of the consumer. In many cases, it is easier to estimate the demand function of a market rather than the individual preferences of all the consumers in the economy. For taxation, it will generally be the case that the magnitude of the tax revenue will be less than the magnitude of the loss in consumer wellbeing. This means that 'taxation is a leaky bucket'. The loss to consumers is larger than the revenue collected by the government. 15. Compensating variation is negative if the price of a good declines. True or False. Answer: True Help: A price decline makes the consumer better off. To offset the increase in utility, the compensating variation must be negative 16. Consumer surplus is zero if demand is perfectly elastic. True or false. Answer: True Help: If demand is perfectly elastic, there is no consumer surplus. CHAPTER 7: Technology and Production Output: the physical products or services that a firm produces Inputs: the materials, labor, land or equipment that a firm uses to produce its output Production Technology: summarizes all of a firm's possible methods for producing its output. - A production method is efficient if there is no other way for the firm to produce a larger amount of output using the same amounts of input - A firm's production possibilities set contains all combinations of inputs and outputs that are possible given the firm’s technology - A firm's efficient production frontier contains the combinations of input and output that a firm can achieve using efficient production methods Production function: a mathematical function giving the amount a firm can produce from given amounts of inputs using efficient production methods - A variable input can be adjusted over the time period being considered, while a fixed input cannot Short Run: period of time over which one or more inputs are fixed Long Run: period of time which all inputs are variables 2 Measures of Productivity: - Average Product of Labor: average output per unit of labor APL = Q/L - Marginal Product of Labor: rate at which output changes when L changes MPL = ΔQ/ΔL The Law of Diminishing Marginal Returns: the general tendency of the marginal product of an input to eventually decline as its use is increased, holding all other inputs fixed Relationship between APL and MPL. WHen the marginal product of an input is larger than/smaller than/the same as the average product, then a marginal increase in the input raises/lowers/does not affect the average product -OR- The APL curve is rising/falling/neither rising nor falling at L if the MPL curve is above/below/equal to the APL curve Production Decisions using MPL: If labor is finely divisible and workers are best assigned to both plants (instead of only one plant) then the marginal product of labor at each plant must be equal. Otherwise, we could do better (increase overall output) by changing the assignment of labor a little bit. Production with Two Variable Inputs Isoquants: identifies all the input combinations that efficiently produce a given amount of output Family of Isoquants: consist of the isoquants corresponding to all of a firm’s possible output levels. The Productive-Inputs Principle: Increasing the amounts of all inputs strictly increases the amount of output the firm can produce (using efficient production methods) Properties of Isoquants - They are thin - Do not slope upward - Is the boundary between the input combinations that produce more and less than a given amount of output - Isoquants for the same technology (and from the same family) do not cross - Isoquants representing higher output levels lie further from the origin Marginal Rate of Technical Substitution: the rate at which a firm must replace units of labor with units of capital to keep output unchanged, starting with a given input combination (when changes involved are tiny). It equals the slope of the firm’s isoquant at this input combination, multiplied by -1. MRTSLK = MPL /MPK Returns to Scale: Tells us about an important property of the production function: Is it profitable to produce more stuff. Or, in other words, do per-unit cost of production go up or down as you produce more stuff? Constant RTS: a proportional change in all inputs produces the same proportional change in output Increasing RTS: a proportional change in all inputs produces a more than proportional change in output Decreasing RTS: a proportional change in all input produce a less than proportional change in output Module on Technology and Production (Chapter 7) Firms produce output using inputs. We generally assume that firms produce efficiently. - Isoquant: Connects the various input combinations with which the firm can produce a particular output level. The MRTSLK indicates the amount of capital that the firm needs to replace one unit of labor if it wants to keep producing the same output level. MRTSLK = MPL /MPK - Isoquants have the same properties as indifference curves - The MP of an input indicates the additional amount of output the firm can produce if it employs one more unit of that input. MP/(input price) gives the additional unit of output that can be produced by spending one more dollar on that input. (Input Price)/MP indicates the additional amount of money needed to produce one more unit of output with that particular input (it is equal to MC at the cost-minimizing input combination in the long run) - The AP of an input indicated the average amount of output produce by one unit of that input E.g., APL = Q/L Relationship: If MP is above/equal to/below AP, then AP is increasing/constant/decreasing MPs for Production Decisions: Allocates workers into plant 1 as long as MP1>MP2. Otherwise, allocate them to plant 2. For ‘continuous’ MPs: 1) MP1=MP2 and 2) L1+L2 = total number of workers. Solve 2) for one of the variables and plug this expression into 1) Different Production Functions Notice: There are no upward sloping isoquants. If a firm wants to reduce one input, it cannot reduce the other input as well and still produces the same output For diagrams of the isoquants of the following production functions, please refer to the previous module on preferences and replace x and y with labor and capital and utility with output. a) Declining MRTS: Isoquants are downward sloping curves. MRTS changes along an isoquant. MRTS = price ratio at cost-minimizing input combination in the long run. The optimal input combination in the long run changes with input prices. b) Perfect Substitutes: Isoquants are straight lines. Constant MRTS. The firm will generally use only one of the two inputs to produce (corner solution), unless MRTS = price ratio. c) Perfect Complements: L-shaped isoquants. The frim necessarily needs to pair a certain number of labor with a certain number of capital. Substitution of inputs is generally not possible. The optimal input combination in the long run is at the corner of the isoquant. The frim does not react not react to a change in input prices if it wants to continue producing the same output level. Returns to Scale Tell us about an important property of the production function: Is it profitable to produce more and more stuff? Or, more precisely, does the per-unit cost of production go up or down as you produce more units of output? (Of course, costs are only one side of the profit-maximization equation; revenue is the other. But this property only talks about the cost side). a) Decreasing RTS: A proportional change in all the inputs produces a less than proportional change in output. E.g. Increase both inputs from 1 to 2. Output increases to less than twice the initial output b) Constant RTS: A proportional change in all the inputs produces the same proportional change in output. E.g: Increase both inputs from 1 to 2. Output increases exactly by a factor 2 c) Increasing RTS: A proportional change in all the inputs produces a more than proportional change in output. E.g.: increase both inputs from 1 to 2. Output increases to more than twice the initial output. L Q Quiz 5: Questions 1. Consider a production technology with the following specification. In 0 0 the table, L is the number of workers and Q is the total amount of output produced by these workers. What is the marginal product of 1 10 worker 4.0? 2 40 Answer: 70 3 90 2. Consider the same company as in the previous question. What is the average product with 5.0 workers? 4 160 Answer: 50 5 250 3. Consider the same firm as question 2 and 3. The production function 6 330 of this firm exhibits decreasing marginal returns for some amount of labor. True or False Answer: True Help: The marginal product of worker 6 is smaller than the marginal product of worker 5. Worker 5 adds 90 units of output but worker 6 only adds 80. 4. If the marginal product of an input is falling at a particular input level, then its average product cannot be rising at that input level. True or False. Answer: False Help: As long as the MP is above the AP, AP is increasing. It does not matter whether MP is increasing or decreasing at that point. 5. Consider the following production function: Q = 25∗L^(½)∗K^(½). Suppose that in the short run, capital is fixed at 36 units. Suppose that the firm wants to produce 300.0 units of output efficiently. How many units of labor does it need? No units, no rounding. Answer: 4 6. The marginal rate of technical substitution tells us the rate at which the firm is able to exchange inputs at prevailing input price ratio, while keeping output fixed. True or False. Answer: False Help: The MRTS is independent of input prices. It tells us at which rate the firm can replace one input with another while still producing the same output level. 7. Suppose that a firm can produce the same output in two different plants. It will allocate all labor to the plant that initially has the highest marginal product of labor. True or False Answer: False Help: The firm will redistribute labor between the two plants until the MP is the same in both plants (unless, of course, one plant is always more efficient than the other) 8. You have an economics test in two weeks. It will cover both micro- and macroeconomics. Each part will be worth 50 points. You have a total of 100 hours that you can spend studying for this exam. For the first 15.0 hours you spend studying microeconomics, you will get one point more for each additional hour of studying. After that, you will get one point more for each three hours you spend studying, up to the maximum possible of 50 points. Macroeconomics is different. For your first 25.0 hours of studying you will get one point for each 45 minutes you spend studying. After that, you will get one point for each 1.5 hours you spend up to the maximum of 50 points. How many hours in total should you spend studying macroeconomics? Answer: 50 Help: First you study macroeconomics because you receive the most points per hour (4/3 points per hour). After that you switch to microeconomics because it gives you the second highest marginal product (1 point per hour). Then you switch back to macroeconomics because you get 2/3 points per hour. You continue studying macroeconomics for the amount of hours needed to get 50 points in total. The remaining hours are spent studying micro again where you now get 1/3 points per hour. 9. Beta Inc. has two production plants and 80.0 workers. The marginal product of labor 1 2 at plant 1 is 𝑀𝑃𝐿=100/L1 and the marginal product of labor at plant 2 is 𝑀𝑃𝐿=100/L2. How many workers will Beta Inc. optimally assign to plant 1? No units, no rounding. Answer: 40 Help: Use the following two conditions to solve this problem. 1) The marginal product at both plants has to be equal. 2) The number of workers in both plants has to add up to the total number of workers. 10. The production function Q = 2K +3L exhibits constant returns to scale. True or false. Answer: True 11. If a dollar spent on labor results in an increase in output that is less than for a dollar spent on capital, a firm can increase output by spending a dollar less on capital and a dollar more on labor. True or false. Answer: False Help: At the cost of minimizing input combination, MPL/MPK=w/r or MPL/w=MPK/r. However, if MPL/w is smaller than MPK/r, then the firm can increase output by spending a dollar more on capital and a dollar less on labor. 12. Suppose that the production function is as follows: Q=min[L,2K]. If the wage rate is $5 and capital costs $10, then the firm will use a different input combination to produce 10 units of output than if the wage rate is $10 and the cost of capital is $5. True or false. Answer: False Help: This is a perfect-complement production function. If the firm wants to produce 10 units of output, it needs 10 units of labor and 5 units of capital, independent of the input prices. 13. Consider the same production function as in the question above. If the firm wants to spend exactly $100 on production and produce the most it can at the current input prices. Then it will use a different input combination if the cost of labor is $5 and the cost of capital is $10 than if the cost of labor is $10 and the cost of capital is $5. True or false. Answer: True Help: At the optimal input combination, we have L=2*K. Therefore, if the firm wants to spend exactly $100, it will have the following cost constraint: 100=w*L+r*K=w*2K+r*K. For given input prices, it will use K=100/(r+2w). This is clearly different for the different price combinations. The firm can produce more output if the wage is cheaper than the cost of capital than if the reverse is true. 14. Consider the following production function: Q = 3K+L. Also, suppose that the wage rate is $2 and capital costs $4. Which of the following statements is true? The firm will only use capital to produce. The firm will only use labor to produce. The firm will use three times as much labor as capital to produce. The firm will use three times as much capital as labor to produce. Answer: The firm will only use capital to produce. Help: This production function is a perfect-substitute production function. The firm is always indifferent between using three units of labor and one unit of capital. However, capital is only twice as expensive. Therefore, the firm will only use capital to produce. 15. The cost of producing output in the long run cannot be greater than the cost of producing the same output in the short run. True or false. Answer: True Help: A firm has more flexibility in the long run than in the short run. Therefore, it cannot be the case that costs in the long run are larger than cost in the short run. CHAPTER 8: Costs Total Cost: of a firm producing a given level of output is the expenditure required to produce that output in the most economical way - Costs are either variable or fixed and fixed costs are either avoidable or sunk/avoidable - Costs that need to be taken into account are opportunity cost, i.e. the cost associated with forgoing the opportunity to employ a resource in its best alternative use. Long-Run Cost Minimization - Isocost Line: contains all the input combinations with the same cost, given the input prices - Family of Isocost Lines: contains, for given input prices, the isocost lines for all of the possible cost levels of the firm 𝑀𝑃𝐿 𝑤 𝑀𝑃𝐿 𝑀𝑃𝐾 Tangency Condition for an interior solution: 𝑀𝑅𝑇𝑆𝐿𝐾 = 𝑀𝑃𝐾 = 𝑟 or 𝑤 = 𝑟 - At a boundary solution at which the cost-minimizing input combination excludes some inputs, the tangency condition might not hold Long-Run Cost Function: tells us how total cost change with output for given input prices Output Expansion Path: shows the least-cost input combinations at all possible levels of output for fixed input prices Average Cost: average cost per unit of output AC=C/Q ∆𝐶 Marginal Cost: rate at which total cost changes when output changes 𝑀𝐶 = ∆𝑄 Efficient Scale of Production: for a firm is the output level at which its average cost is the lowest. - AC and MC cross at the efficient scale of production 𝑤 𝑟 Marginal Products, Marginal Costs and Input Prices: 𝑀𝐶 = 𝑀𝑃𝐿 = 𝑀𝑃𝐾 Effects of Input Price Changes: generally, when the price of an input decreases, the firm’s least-cost production method never uses less of that input, and usually employs more. When the price of an input increases, a firm’s least-cost production method never uses more of that input, and usually employs less Economies of Scale and Returns to Scale: a firm experiences economies of scale when its average cost falls as it produces more, and diseconomies of scale when its average cost rises as it produces more - Increasing RTS lead to economies of scale - Decreasing RTS lead to diseconomies of scale MODULE - Cost (Ch. 8) When firms produce, they must pay for the inputs. The cost of labor is w and the cost of capital is r. Cost functions have the form: C = w*L + r*K Suppose the firm wants to produce Q =100. In the Short run, K is fixed, so the input combination is (KFixed, LSR). In the long run, both K and L are flexible, sothe optimal input combination is (L*, K*). Often, the input is not fixed at an optimal level (see above diagram). Then CSR> CLR. Short Run: Put fixed input in the production function and figure out how many units of the other input are required to produce the desired output level. Calculate cost with the above formula. Long Run: Find the optimal input combination (Hint: The recipes from consumer choice can be applied here. Please refer to the respective video tutorial and review module. The only difference is that the second step consumer choice (budget line) is replaced either with the production function (fixed output) or the cost line (fixed cost/budget).) a) Declining MRTS: 2 conditions to find the cost minimizing input combination to produce a certain output level. 1. MRTSLK= w/r. Solve for one of the inputs. 2. Plug the expression from 1) into the production function. Sometimes, not the output level but the expenditure (cost) is the constraint. If the firm wants to spend only a certain amount of money, then plug the expression from 1) into the cost function. When input prices change, the firm will adjust its input combination to produce a given level of output b) Perfect Substitutes: The firm will use only one input unless MRTSLK= w/r. (Refer to the Module on Consumer Choice for the optimal input combination depending on the input price ratio.) Figure out how much of that input is required to produce the required output level. Calculate cost with the above formula. When input prices change,the firm may or may not adjust its input combination if it wants to continue producing a certain output level. It depends on the relative prices after the price change. c) Perfect Complements: The firm will use the two inputs in a fixed ratio (refer to the Module on Consumer Choice.) Figure out how many units of the two inputs are required to produce the desired output level. Calculate cost with the above formula. When input prices change, the firm cannot adjust its input combination if it wants to continue producing a certain output. Long Run and Short Run Cost Functions: - Short run costs are at least as large as long run costs. They have one point in common (the output level at which the input is fixed at the optimal level). - The same relationships hold for other cost functions (short run and long run AC, short run and long run MC...) Important Cost Functions: - QE is the efficient scale of production. It is the output level at which average costs are minimized. Existing firms produce this output level in the long run competitive market equilibrium. If the market price is equal to min AC, firms make zero profit MC intercepts all of the following cost functions at their minimum. If you want to calculate the minimum, set the respective cost function equal to MC. - AC = total cost divided by Q (includes all cost) - AAC = variable cost divided by Q (excludes sunk costs) - AVC = variable costs divide by Q (excludes sunk costs and avoidable fixed costs) Economies of Scale A firm experiences economies of scale if its average cost fall as it produces more output; it experiences diseconomies of scale if its average cost increases as it produces more output. - Increasing RTS ( a property of the production function) lead to economies of scale - Decreasing RTS lead to diseconomies of scale Quiz 6 Questions 1. If marginal cost is equal to average variable cost at a particular output level and average variable cost are not the same as average cost (i.e., there are fixed cost), then the average cost curve is downward sloping at that output level. True or false. Answer: True Help: The statement of this question implies that the minimum of the AC curve must be to the right of the minimum of the AVC curve. Why is that? Suppose that we are at the output level at which AVC are minimized (the output level at which MC intersects with AVC, i.e., they are the same). Also assume that the output level at which AC is minimized is smaller. This implies that there is a region of output levels for which AC is increasing while AVC is decreasing. This cannot be the case. Why? Remember that AC=AVC+AFC. For sure, AFC is decreasing with output. If AVC is also decreasing, it cannot be the case that AC is increasing. In conclusion: The output level at which AVC is minimized is always smaller than the output level at which AC is minimized (unless there is no fixed cost and the two curves coincide). 2. The following relationship must hold between the average cost curve (AC) and the marginal cost curve (MC): Answer: If AC is rising, MC must be greater than AC 3. Suppose that in the short run a firm has fixed cost of 54 and variable cost of 1.5Q2. Short-run marginal cost are MC=3Q. What are short-run average cost? Answer: AC(Q) = 54/Q + (3/2)Q 4. Consider the same cost as in the question above. The minimum of the short-run average cost is at ____ units of output. At this point, short-run average costs are equal to ____. Answer: 6;18 Help: Set MC equal to AC to calculate the output level at which AC is minimized. Plug this output level in either AC or MC (it does not matter which because they are equal at that output level) to calculate AC at that output level. 5. Suppose that a firm's production function exhibits a declining MTRS. Then marginal costs in the long run are positively related to the marginal product of capital, i.e., if the marginal product increases, marginal cost increases as well. True or false. Answer: False Help: First of all, the equation at the optimal input combination in the long run is w/MPL=r/MPK=MC. Therefore, MC decreases when MP increases.Intuitively: Remember that marginal cost are the additional cost incurred when producing one more unit of output. Therefore, if the additional output per extra unit of labor increases, i.e., MPL increases, marginal cost per unit of output must decrease. The same argument applies to capital. 6. Suppose that a firm uses yinks and gorks to produce high-quality zurbs. Suppose that the isoquants have a declining MRTSYG. At the current input combination, the marginal product of yinks is 10. The price of a yink is $2 and the price of a gork is $1.0. What must the marginal product of gorks be at the current input combination if the firm cannot produce the same output at a lower cost by re-adjusting inputs? No units, no rounding. Answer: 5 Help: Notice that the question describes a firm that uses an optimal input combination. For a production function with declining MRTS, it has to be true at the cost-minimizing input combination that MPL/w = MPK/r. Apply this formula to the present inputs and you can solve for the missing marginal product. 7. Suppose a firm as the following production function: Q = 4∗L^(0.5)∗K^(0.5) = 4∗√L∗√K. Suppose that the cost of capital is $200 and the wage rate is $2000. The firm has a fixed capital input of 25 units. What is the firm's short-run cost function? 𝑆𝑅 2 Answer: 𝐶 (𝑄) = 5000 + 5𝑄 Help: With the fixed capital, the production function becomes Q = 20∗√L. Therefore, the labor demand of this firm is equal to L∗ = Q^2/400. The firm spends $5000 on the fixed capital and 2000L* on labor. Plug these into the cost function. 8. Pat’s company, Tasty Tidbits, competes with Evie’s and Johnaas’s company in the Guelph catering market. His production function is: Q = 20 * √L * √K with MRTSLK = K/L. Pat pays a fair wage of $225 and a unit of capital cost $100. Currently, Pat has a fixed stock of 25.0 units of capital. How much could Pat save when producing 900.0 meals (Q = 900.0) efficiently if the capital stock was not fixed? If Pat could save money, then enter a positive amount. Answer: 7225 Help: A Cobb-Douglas production function is of the form Q = A∗L^α∗K^β where A, α, β are positive numbers. For these production function, MRTSLK=α/β∗K/L. For the way to calculate the cost minimizing input combination and associated total cost in the short run, please refer to the feedback in the previous question. How to calculate the cost-minimizing input combination in the long run? Suppose, for example, that the firm wants to produce Q=900. 1) MRTSLK=K/L=w/r=225/100. Therefore, K=2.25*L. 2) Q=900=20*L0.5*K0.5. Plug the above K into this equation. Then you get 900=20*1.5*L and therefore L=30 and K=67.5. Given the input prices, you can now calculate the total cost of production. You will find that Pat spends less money if the input is not fixed. 9. Consider the same production function as in the question above. Suppose that Pat wants to produce 600.0 meals and no input is fixed. The price of capital is $16. Initially, the price of labor is $16, but then the price of labor increases to $36. How much more does Pat have to pay when the price of labor increases if he still wants to produce 600.0 meals? Answer: 480 Help: For the way to calculate the cost-minimizing input combination and associated cost in the long run, please refer to the feedback in the previous question. 10. Johnson Tools produces hammers. Suppose that university graduates earn $25 per hour and high school graduates earn $15 per hour. Suppose that the marginal product of a university graduate at Johnson Tools is five hammers per hour and that the marginal product of high school graduates is four hammers per hour (regardless of the number of each type of worker employed). Which of the following statements is true if Johnson Tools wants to produce 90 hammers? Answer: Johnson Tools will employ only high school graduates Help: Clearly, this is a perfect substitutes production function. A hammer produced by a university graduate always costs 25/5=5 and a hammer produced by a high school graduate always costs 15/4=3.75. High school graduates produce hammers more cheaply and therefore Johnson Tools only employs high school graduates. 11. Consider the same production function as above. Johnson Tools still wants to produce the same amount of output. Question 1: Will Johnson Tools use a different input combination if the price of high school graduates increases to $19? Question 2: Will Johnson Tools use a different input combination if the price of high school graduates increases to $21? Answer: No; yes Help: As long as the wage of high school graduates is below $20, Johnson Tools hires only high school graduates. If the wage of high school graduates is larger than $20, Johnson Tools hires only university graduates. If the wage of high school graduates is $20, Johnson Tools is indifferent and may hire both types or only one type of worker. 12. Suppose that a company uses a perfect substitutes production function and wants to produce 100 units of output with two inputs regardless of cost. If the price of one of the two inputs decreases, the company will necessarily use more of that input than before. True or false. Answer: False Help: False. Recall that this is a perfect substitute production function. The firm may continue using only the other input. Even though the price of one input decreases, this input may still be the relatively more expensive one. Refer to the questions above involving Johnson Tools. Notice that the answer would be different if the production function exhibited a declining MRTS. Then a decrease in an input price would ALWAYS lead to more use of that input. 13. Suppose that a firm uses a perfect complements production function and wants to produce 100 units with two inputs regardless of cost. Suppose that the price of one of the two inputs decreases. The firm will use more of that input than before. True or false. Answer: False Help: If a firm uses a perfect complements production function, it needs to employ the inputs in a fixed proportion. It cannot change the input combination if it wants to continue producing the same amount of output, regardless of input prices. Notice that the answer would be different for perfect substitutes and declining MRTS technologies. For these, please refer to the feedback in Question 12. 14. If the total cost for a firm increases at the same rate as output, then this firm must use a technology that exhibits decreasing returns to scale. True or false. Answer: False Feedback: This is false. If the cost increases at the same rate as output, then the firm's production function exhibits constant returns to scale. In this case, average costs are constant. Average costs are constant only if the firm exhibits constant returns to scale. Generally: If a firm's average cost is decreasing, we say that it exhibits 'economies of scale'. Increasing returns to scale lead to economies of scale. If a firm's average cost is increasing, we say that it exhibits 'diseconomies of scale'. Decreasing returns to scale lead to diseconomies of scale. If a firm's average costs are neither increasing nor decreasing, they are constant. This happens when the firm has constant returns to scale. CHAPTER 9: Profit Maximization Profit: of a firm, π, is equal to revenue R less cost C - The firm’s profit maximization problem is to find the quantity and/or price that results in the largest possible profit. Feasible pairs of price and quantity can be found along the demand function What happens when a firm changes price/quantity along a demand curve? Suppose Q increases: 1) Output Expansion Effect: ΔQ additional units are sold, which increases R 2) Price Reduction Effect: The increase in Q requires that the price is reduced for all inframarginal units, which decrease R A firm will only increase its quantity if revenue increases and if the additional revenue (MR) is larger than the additional cost (MC) - Marginal Revenue: the rate at which R increases when one extra marginal unit of Q is sold. MR = ΔR/ΔQ The Two-Step Method for a Price Taking Firm 1) Quantity Rule: Identify all possible sales quantities at which MR=MC. If there are several determine which produces the highest profit 2) Shut-Down Rule: Check whether the most profitable positive sales quantity from Step 1 results in greater profit than shutting down. If it does, that is the best choice. Otherwise, shutting down is the best choice. IF they are the same, then either choice maximizes profit. Shut-Down Rule for a Price-Taking Firm - If the market price P* is at least as large as minimum average avoidable cost AACmin, then produce the quantity at which P* = MC. If P* < AACmin, they shut down and produce nothing (profit may be negative because of sunk cost) Producer Surplus: the difference between the amount the firm receives from selling the good, the price and the minimum amount the frim must receive in order to supply the good. - Producer Surplus (PS) = Revenue - Avoidable Cost - Profit = PS - Sunk Cost Module on Profit Maximization (Ch. 9) - Every firm that produces positive quantities sets MR = MC - For firms without market power (i.e. firms in a perfectly competitive market), MR = P* (market price), so P* = MC - Firms in a perfectly competitive market only supply positive quantities is P* ≥ PSD (the shutdown price). PSD = min AAC Consider the diagram. If the market price is P*, this firm produces a positive quantity because P* is larger than the firm’s shut-down price PSD. It produces a quantity at which P*=MC. Th

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